Heegner Points

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$, and fix a modular parametrization $\pi_E:X_0(N) \to E$.

Let $K$ be a quadratic imaginary field such that the primes dividing $N$ are all unramified and split in $K$. For simplicity, we will also assume that $K\neq \mathbb{Q}(i), \mathbb{Q}(\sqrt{-3})$. Let $\mathcal{N}$ be an integral ideal of $\O_K$ such that $\O_K/\mathcal{N}\cong \mathbb{Z}/N\mathbb{Z}$. Then $\mathbb{C}/\O_K$ and $\mathbb{C}/\mathcal{N}^{-1}$ define two elliptic curves over $\mathbb{C}$, and since $\O_K \subset \mathcal{N}^{-1}$, there is a natural map

$$\mathbb{C}/\O_K \to \mathbb{C}/\mathcal{N}^{-1}.$$ (3.2.1)

By Proposition 3.3 the kernel of this map is

$$\mathcal{N}^{-1}/\O_K \cong \O_K/\mathcal{N}\cong \mathbb{Z}/N\mathbb{Z}.$$

Exercise 3.19   Prove that there is an isomorphism $\mathcal{N}^{-1}/\O_K \cong \O_K/\mathcal{N}$ of finite abelian group.

The modular curve $X_0(N)$ parametrizes isomorphism classes of pairs $(F,\phi)$, where $\phi$ is an isogeny with kernel cyclic of order $N$. Thus $\mathbb{C}/\O_K$ and the isogeny (3.2.1) define an element $x_1 \in X_0(N)(\mathbb{C})$. The discussion of Section 3.1.3 along with properties of modular curves proves the following proposition.

Proposition 3.20   We have

$$x_1 \in X_0(N)(H),$$

where $H$ is the Hilbert class field of $K$.

Definition 3.21 (Heegner point)   The Heegner point associated to $K$ is

$$y_K = \Tr _{H/K}(\pi_E(x_1)) \in E(K).$$

More generally, for any integer $n$, let $\O_n = \mathbb{Z}+ n\O_K$ be the order in $\O_K$ of index $n$. Then $\mathcal{N}_n = \mathcal{N}\cap \O_n$ satisfies $\O_n/\mathcal{N}_n \cong \mathbb{Z}/N\mathbb{Z}$, and the pair

$$(\mathbb{C}/\O_n, \mathbb{C}/\O_n \to \mathbb{C}/\mathcal{N}_n^{-1})$$

defines a point $x_n \in X_0(N)(K_n)$, where $K_n$ is the ray class field of conductor $n$ over $K$.

Definition 3.22 (Heegner point of conductor $n$)   The Heegner point of conductor $n$ is

$$y_n = \pi_E(x_n) \in E(K_n).$$